Cylinder Radius Calculation A Step-by-Step Guide

Hey there, math enthusiasts! Today, let's dive into the fascinating world of cylinders and explore how to calculate their radii. We've got a cylinder with a volume of 4πx³ cubic units and a height of x units. Our mission, should we choose to accept it, is to find the expression that represents the radius of this cylindrical wonder. So, buckle up, grab your thinking caps, and let's embark on this mathematical adventure together!

Unraveling the Formula The Key to Cylindrical Calculations

Before we jump into the nitty-gritty of our specific problem, let's refresh our understanding of the fundamental formula for the volume of a cylinder. Remember, the volume of a cylinder is the amount of space it occupies, and it's calculated using a simple yet elegant formula: Volume = πr²h, where:

• Volume is the total space enclosed by the cylinder.
• π (pi) is a mathematical constant approximately equal to 3.14159.
• r is the radius of the circular base of the cylinder, which is the distance from the center of the circle to any point on its edge.
• h is the height of the cylinder, which is the perpendicular distance between the two circular bases.

This formula is the cornerstone of our cylindrical calculations, and it's the key to unlocking the mystery of the radius in our problem. So, let's keep this formula etched in our minds as we move forward.

Now that we've got our formula in hand, let's take a closer look at the information we've been given. We know that the volume of our cylinder is 4πx³ cubic units, and its height is x units. Our goal is to find the radius, which is represented by the variable r in our formula. To do this, we'll need to manipulate our volume formula to isolate r on one side of the equation. This will involve some algebraic maneuvering, but don't worry, we'll take it step by step and make sure everything is crystal clear. With a little bit of algebraic magic, we'll be able to unveil the expression for the radius of our cylinder. So, let's roll up our sleeves and get ready to solve for r!

Applying the Formula The First Step Towards Finding the Radius

Now comes the exciting part where we put our formula to work! We know the volume and the height, and we want to find the radius. So, let's plug the given values into our trusty volume formula: Volume = πr²h.

We're told that the volume is 4πx³ cubic units, so we can substitute that in for Volume. We also know that the height is x units, so we'll substitute that in for h. This gives us the equation:

4πx³ = πr²x

See how we've replaced the general terms Volume and h with the specific values we were given? This is a crucial step in solving for the radius. Now, we've got an equation that relates the radius, r, to the known quantities. Our next task is to manipulate this equation to isolate r² on one side. This will bring us one step closer to finding the radius itself. So, let's put on our algebraic hats and get ready to simplify!

Isolating the Radius The Algebraic Dance

Our goal is to get r² all by itself on one side of the equation. To do this, we need to perform some algebraic operations. Remember, whatever we do to one side of the equation, we must also do to the other side to maintain the balance.

Looking at our equation, 4πx³ = πr²x, we can see that r² is being multiplied by both π and x. To get rid of these, we'll need to divide both sides of the equation by πx. This is the key move that will start to isolate r². When we divide both sides by πx, we're essentially undoing the multiplication that's happening on the right side of the equation. This will leave us with r² by itself, which is exactly what we want.

So, let's perform this division. Dividing both sides of 4πx³ = πr²x by πx, we get:

(4πx³)/(πx) = (πr²x)/(πx)

Now, let's simplify this expression. On the left side, the π in the numerator and denominator cancel out, and we can simplify x³/x to x² (remember, when dividing exponents with the same base, you subtract the powers). On the right side, both the π and the x in the numerator and denominator cancel out, leaving us with just r². This is exactly the simplification we were hoping for!

This simplifies to:

4x² = r²

We've successfully isolated r²! We're almost there. Now, we just need to take one more step to find r itself. Are you ready for the final flourish? Let's do it!

Finding the Radius The Grand Finale

We've got 4x² = r², and we want to find r. The key here is to realize that r is the square root of r². So, to find r, we need to take the square root of both sides of the equation. This is the final step in our algebraic journey, and it will reveal the expression for the radius of our cylinder.

Taking the square root of both sides of 4x² = r², we get:

√(4x²) = √r²

Now, let's simplify this. The square root of 4x² is 2x (since √(4) = 2 and √(x²) = x), and the square root of r² is simply r. This gives us:

2x = r

And there you have it! We've found the expression for the radius of the cylinder. The radius, r, is equal to 2x units. This is the solution we've been searching for, and it's the answer to our mathematical quest.

Conclusion Decoding Cylindrical Dimensions

Woo-hoo! We did it, guys! We successfully navigated the world of cylinders, deciphered the volume formula, and found the expression for the radius. It's like we're mathematical superheroes, armed with the power of algebra and geometric knowledge. Give yourselves a pat on the back – you've earned it!

In this problem, we started with a cylinder with a given volume and height, and our mission was to find the radius. We used the formula for the volume of a cylinder, Volume = πr²h, as our trusty guide. We plugged in the given values, performed some algebraic gymnastics to isolate r, and finally arrived at the answer: the radius of the cylinder is 2x units.

This exercise not only helped us solve a specific problem but also reinforced our understanding of cylindrical geometry and algebraic manipulation. These are valuable skills that can be applied to a wide range of mathematical challenges. So, the next time you encounter a cylinder, you'll be ready to tackle it with confidence and skill!

And remember, mathematics is not just about numbers and formulas; it's about problem-solving, critical thinking, and the joy of discovery. So, keep exploring, keep questioning, and keep unlocking the secrets of the mathematical universe. Until next time, happy calculating!

So, the correct answer is A. 2x. We hope you enjoyed this mathematical adventure, and we look forward to exploring more fascinating concepts with you in the future!